Cesàro summability of one- and two-dimensional Walsh-Fourier series

Weisz, Ferenc

doi:10.1007/bf02205221

Cited by 113 publications

(98 citation statements)

References 4 publications

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“…also [2]) for bounded Vilenkin series. Weisz [20] proved that the maximal operator of the Fejér means σ * is bounded from the Hardy space H 1/2 to the space weak-L 1/2 . Simon [12] gave a counterexample, which shows that boundedness does not hold for 0 < p < 1/2.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

A note on the maximal operators of Vilenkin—Nörlund means with non-increasing coefficients

Memić

Persson

Tephnadze

2016

Studia Scientiarum Mathematicarum Hungarica

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In [14] we investigated some Vilenkin-Nörlund means with non-increasing coefficients. In particular, it was proved that under some special conditions the maximal operators of such summabily methods are bounded from the Hardy space H 1/(1+α) to the space weak -L 1/(1+α) , (0 < α 1). In this paper we construct a martingale in the space H 1/(1+α) , which satisfies the conditions considered in [14], and so that the maximal operators of these Vilenkin-Nörlund means with non-increasing coefficients are not bounded from the Hardy space H 1/(1+α) to the space L 1/(1+α) . In particular, this shows that the conditions under which the result in [14] is proved are in a sense sharp. Moreover, as further applications, some well-known and new results are pointed out.

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

A note on the maximal operators of Vilenkin—Nörlund means with non-increasing coefficients

Memić

Persson

Tephnadze

2016

Studia Scientiarum Mathematicarum Hungarica

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“…But, Fujii [3] proved that σ * is bounded from the dyadic Hardy space H 1 to the space L 1 . The theorem of Fujii was extended by Weisz [21], he showed that the maximal operator σ * is bounded from the martingale Hardy space H p to the space L p for p > 1/2. Simon gave a counterexample [13], which shows that the boundedness does not hold for 0 < p < 1/2.…”

Section: Introductionmentioning

confidence: 99%

On the maximal operator of Walsh-Kaczmarz-Fejér means

Гогинава

Nagy

2011

Czech Math J

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“…Namely, for functions in L 1 (I 2 ) the a.e. convergence σ (n1,n2) f → f holds as the indices n 1 , n 2 → ∞ restricted as β −1 ≤ n 1 /n 2 ≤ β for some β > 1 (compare with [7], [8], [9]). …”

Section: The Construction Of the Functionmentioning

confidence: 99%

On the divergence of the $(C,1)$ means of double Walsh-Fourier series

Gát¹

1999

Proc. Amer. Math. Soc.

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Abstract. In 1992, Móricz, Schipp and Wade proved the a.e. convergence of the double (C, 1) means of the Walsh-Fourier series σnf → f (min(n 1 , n 2 ) → ∞, n = (n 1 , n 2 ) ∈ N 2 ) for functions in Llog + L(I 2 ) (I 2 is the unit square). This paper aims to demonstrate the sharpness of this result. Namely, we prove that for all measurable function δ : [0, +∞) → [0, +∞), limt→∞ δ(t) = 0 we have a function f such as f ∈ Llog + Lδ(L) and σnf does not converge to f a.e. (in the Pringsheim sense).

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Cesàro summability of one- and two-dimensional Walsh-Fourier series

Cited by 113 publications

References 4 publications

A note on the maximal operators of Vilenkin—Nörlund means with non-increasing coefficients

A note on the maximal operators of Vilenkin—Nörlund means with non-increasing coefficients

On the maximal operator of Walsh-Kaczmarz-Fejér means

On the divergence of the $(C,1)$ means of double Walsh-Fourier series

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